The exemplary embodiment relates to a method and system of selling goods and services. It finds particular application in connection with methods for learning multidimensional pricing mechanisms.
A buyer considering a purchase is likely to buy an item if the value that he places on the item is greater than the price he will have to pay for the item. However, buyers rarely value only one alternative when considering a purchase. There may be near-substitute products (e.g., multiple TVs), alternative payment methods (e.g., corresponding to different buyer time discounting or interest rates), varying quantities and qualities of the same product, or different expectations about mean future values (e.g., when advance-selling the same item, which can result in arbitrarily more profit than selling one item at a time). All of these factors can affect whether or not a buyer will purchase a given item. From the seller's perspective, each potential buyer has his own value placed on an item, which the seller would like to know in order to establish prices. He may attempt to estimate the buyer's value from prior sales on the assumption that if a prior buyer has paid the price, the buyer's value must have been higher. Sellers generally want to maximize their profit, which is a function of the sales price and the number of goods sold. The seller may adjust the price over time to obtain a better idea of what buyers are willing to pay, but this process can be time consuming and expensive for the seller. Frequent manipulation of pricing can also be problematic, since it can annoy buyer If the price is set too high then buyers may go elsewhere, and may not return to the seller even if the seller later lowers the price.
For purposes of the embodiments described herein, a buyer is described by a point in value space at a particular time, where each axis corresponds to the valuation for one alternative item. For example, where two different items are being sold, each item has its own value axis and the value space is dimensional. A seller's belief about buyers can be considered as a probability density over the value space. The seller's objective is to divide the value space into regions that are served by different contracts. These regions are known as (market) segments.
Optimal pricing mechanisms have been characterized in detail via a function that describes a buyer's utility when they purchase the best possible contract for themselves. This function is known as the mechanism function (see, Mane A., Vincent D. Pricing mechanism design: Revenue maximization and the multiple-good monopoly. J. Economic Theory, 137:153-185 (2007)). Briest, et al. investigated the profit from an optimal lottery relative to the profit from an optimal non-lottery pricing scheme. They showed that the gain is three in two dimensions, and unbounded in four and higher dimensions (Briest, P., Chawla, S., Kleinberg, R., Weinberg, S., Pricing Randomized Allocations. Proc. 21st Annual ACM-SIAM Symp. on Discrete Algorithms (Ed., Moses Charikar, January 2010)). Surprisingly, optimal lotteries may be found efficiently using linear programming or semi-definite programming. Effective methods for solving such problems are discussed by Aguilera and Morin (Aguilera, N., Morin, P., On convex functions and the finite element method. SIAM Journal on Numerical Analysis 47(4):3139-3157, (2009)).
Existing work on pricing tends to focus on setting a price where there is limited supply, e.g., a single item. There remains a need for a method for learning pricing mechanisms for setting prices.